First, I had to prove that $\mathbb{Z}$ $(\sqrt{-17})$ is not a UFD which was done by showing that in general $\mathbb{Z}$ $(\sqrt{-n})$ for $n \geq 3$ is not a UFD. Subsequently, I was asked if (3) is a prime ideal of $\mathbb{Z}$ $(\sqrt{-17})$.
Is it enough to say that 3 divides the product $(1+\sqrt{-17})(1-\sqrt{-17})=18$ but does not divide either of the factors thus it is not a prime ideal?
Is there another way of showing it using, perchance, the UFD definition?